83 Search Results for "Arora, Sanjeev"

Document
DOI: 10.4230/LIPIcs.SWAT.2026.14
Orthogonal Strip Partitioning of Polygons: Lattice-Theoretic Algorithms and Lower Bounds

Authors: Jaehoon Chung

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
We study a variant of a polygon partition problem, introduced by Chung, Iwama, Liao, and Ahn [ISAAC'25]. Given orthogonal unit vectors 𝐮,𝐯 ∈ ℝ² and a polygon P with n vertices, we partition P into connected pieces by cuts parallel to 𝐯 such that each resulting subpolygon has width at most one in direction 𝐮. We consider the value version, which asks for the minimum number of strips, and the reporting version, which outputs a compact encoding of the cuts in an optimal strip partition. We give efficient algorithms and lower bounds for both versions on three classes of polygons of increasing generality: convex, simple, and self-overlapping. For convex polygons, we solve the value version in O(log n) time and the reporting version in O(h log (1 + n/h)) time, where h is the width of P in direction 𝐮. We prove matching lower bounds in the decision-tree model, showing that the reporting algorithm is input-sensitive optimal with respect to h. For simple polygons, we present O(n log n)-time, O(n)-space algorithms for both versions and prove an Ω(n) lower bound. For self-overlapping polygons, we extend the approach for simple polygons to obtain O(n log n)-time, O(n)-space algorithms for both versions, and we prove a matching Ω(n log n) lower bound in the algebraic computation-tree model via a reduction from the δ-closeness problem. Our approach relies on a lattice-theoretic formulation of the problem. We represent strip partitions as antichains of intervals in the Clarke-Cormack-Burkowski lattice, originally developed for minimal-interval semantics in information retrieval. Within this lattice framework, we design a dynamic programming algorithm that uses the lattice operations of meet and join. To the best of our knowledge, this is the first geometric application of the Clarke-Cormack-Burkowski lattice.
Cite as

Jaehoon Chung. Orthogonal Strip Partitioning of Polygons: Lattice-Theoretic Algorithms and Lower Bounds. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chung:LIPIcs.SWAT.2026.14,
  author =	{Chung, Jaehoon},
  title =	{{Orthogonal Strip Partitioning of Polygons: Lattice-Theoretic Algorithms and Lower Bounds}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.14},
  URN =		{urn:nbn:de:0030-drops-260506},
  doi =		{10.4230/LIPIcs.SWAT.2026.14},
  annote =	{Keywords: Polygon partitioning, Strip partition, Lattice, Self-overlapping curves}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2026.22
Search-Space Reduction for Boolean MinCSPs via Essential Constraints

Authors: Bart M. P. Jansen and Ruben F. A. Verhaegh

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
For a fixed set ℱ of Boolean constraint types, a MinCSP(ℱ)-instance consists of a formula F that applies m constraints from ℱ to a set of n Boolean variables. The goal is to remove a minimum subset of constraint applications from F to make the remaining formula satisfiable. Previous work characterized how the choice of ℱ affects its polynomial-time solvability and approximability. We extend a recently introduced preprocessing framework for graph problems to the problem above. Rephrased in the context of CSPs, this framework defines a constraint application from a given formula F as c-essential if it is contained in all c-approximate solutions to F. Being able to efficiently detect these essential parts of a solution reduces the search space of any follow-up FPT algorithms parameterized by the solution size and yields an immediate asymptotic improvement to the runtime of such algorithms. In this work, we present a dichotomy theorem that distinguishes constraint sets ℱ for which c_ℱ-essential constraint applications can be detected efficiently for some c_{ℱ} ∈ 𝒪(1), from those for which this task is intractable under established complexity-theoretic conjectures. Our results show that for any set ℱ of bijunctive constraints, there is a polynomial-time algorithm that detects 𝒪(1)-essential constraint applications. This contrasts the fact that constant-factor approximating a bijunctive MinCSP(ℱ)-problem is intractable under the Unique Games Conjecture.
Cite as

Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-Space Reduction for Boolean MinCSPs via Essential Constraints. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jansen_et_al:LIPIcs.SWAT.2026.22,
  author =	{Jansen, Bart M. P. and Verhaegh, Ruben F. A.},
  title =	{{Search-Space Reduction for Boolean MinCSPs via Essential Constraints}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.22},
  URN =		{urn:nbn:de:0030-drops-260586},
  doi =		{10.4230/LIPIcs.SWAT.2026.22},
  annote =	{Keywords: fixed-parameter tractability, constraint satisfaction problems}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2026.18
On Fréchet Traveling Salesmen Problems

Authors: Omrit Filtser, Tzalik Maimon, and Michal Moiseev

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
The Fréchet distance is a well-studied distance measure between two curves. In this work, we demonstrate that the merit of Fréchet distance extends beyond evaluating similarity, and introduce a new setting in which it proves useful. Consider a situation where two agents are required to visit a given set of sites, while staying close to each other throughout their traversal. In this paper, we study problems where the goal is to construct two curves whose vertices are from a given set of points, under the constraint that the Fréchet distance between the curves is kept as small as possible. This problem can be viewed as a variant of the Traveling Salesman Problem (TSP), and thus may be of interest in routing, network planning and more. We present a near-linear algorithm for this problem under the discrete Fréchet distance, and explore several variants of the problem, including minimizing the lengths of the curves and balancing the number of sites assigned to each agent. Lastly, we prove that the problem is NP-hard under the continuous Fréchet Distance.
Cite as

Omrit Filtser, Tzalik Maimon, and Michal Moiseev. On Fréchet Traveling Salesmen Problems. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{filtser_et_al:LIPIcs.SWAT.2026.18,
  author =	{Filtser, Omrit and Maimon, Tzalik and Moiseev, Michal},
  title =	{{On Fr\'{e}chet Traveling Salesmen Problems}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.18},
  URN =		{urn:nbn:de:0030-drops-260545},
  doi =		{10.4230/LIPIcs.SWAT.2026.18},
  annote =	{Keywords: Fr\'{e}chet distance, traveling salesman problem}
}
Document
DOI: 10.4230/LIPIcs.FORC.2026.1
Computational Hardness of Private Coreset

Authors: Badih Ghazi, Cristóbal Guzmán, Pritish Kamath, Alexander Knop, Ravi Kumar, and Pasin Manurangsi

Published in: LIPIcs, Volume 368, 7th Symposium on Foundations of Responsible Computing (FORC 2026)

Abstract
We study the problem of differentially private (DP) computation of coreset for the k-means objective. For a given input set of points, a coreset is another set of points such that the k-means objective for any candidate solution is preserved up to a multiplicative (1 ± α) factor (and some additive factor). We prove the first computational lower bounds for this problem. Specifically, assuming the existence of one-way functions, we show that no polynomial-time (ε, 1/n^{ω(1)})-DP algorithm can compute a coreset for k-means in the 𝓁_∞-metric for some constant α > 0 (and some constant additive factor), even for k = 3. For k-means in the Euclidean metric, we show a similar result but only for α = Θ(1/d²), where d is the dimension.
Cite as

Badih Ghazi, Cristóbal Guzmán, Pritish Kamath, Alexander Knop, Ravi Kumar, and Pasin Manurangsi. Computational Hardness of Private Coreset. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ghazi_et_al:LIPIcs.FORC.2026.1,
  author =	{Ghazi, Badih and Guzm\'{a}n, Crist\'{o}bal and Kamath, Pritish and Knop, Alexander and Kumar, Ravi and Manurangsi, Pasin},
  title =	{{Computational Hardness of Private Coreset}},
  booktitle =	{7th Symposium on Foundations of Responsible Computing (FORC 2026)},
  pages =	{1:1--1:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-419-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{368},
  editor =	{Lin, Huijia (Rachel)},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2026.1},
  URN =		{urn:nbn:de:0030-drops-259725},
  doi =		{10.4230/LIPIcs.FORC.2026.1},
  annote =	{Keywords: Differentially Private Clustering, Coreset, Cryptographic Hardness}
}
Document
DOI: 10.4230/LIPIcs.FORC.2026.14
Fair Multi-Agent Persuasion with Submodular Constraints

Authors: Yannan Bai, Kamesh Munagala, Yiheng Shen, and Davidson Zhu

Published in: LIPIcs, Volume 368, 7th Symposium on Foundations of Responsible Computing (FORC 2026)

Abstract
We study the problem of selection in the context of Bayesian persuasion. We are given multiple agents with hidden values (or quality scores), to whom resources must be allocated by a welfare-maximizing decision-maker. An intermediary with knowledge of the agents' values seeks to influence the outcome of the selection by designing informative signals and providing tie-breaking policies, so that when the receiver maximizes welfare over the resulting posteriors, the expected utilities of the agents (where utility is defined as allocation times value) achieve certain fairness properties. The fairness measure we will use is majorization, which simultaneously approximately maximizes all symmetric, monotone, concave functions of the utilities. We consider the general setting where the allocation to the agents needs to respect arbitrary submodular constraints, as given by the corresponding polymatroid. We present a signaling policy that achieves a logarithmically approximate majorized policy in this setting, assuming the receiver is a (1+ε) approximate welfare maximizer. The approximation ratio is almost best possible, and that significantly outperforms generic results that only yield linear approximations. A key component of our result is a structural characterization showing that the vector of agent utilities for a given signaling policy defines the base polytope of a different polymatroid, a result that may be of independent interest. In addition, we show that an arbitrarily good additive approximation to this vector can be produced in (weakly) polynomial time via the multiplicative weights update method.
Cite as

Yannan Bai, Kamesh Munagala, Yiheng Shen, and Davidson Zhu. Fair Multi-Agent Persuasion with Submodular Constraints. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bai_et_al:LIPIcs.FORC.2026.14,
  author =	{Bai, Yannan and Munagala, Kamesh and Shen, Yiheng and Zhu, Davidson},
  title =	{{Fair Multi-Agent Persuasion with Submodular Constraints}},
  booktitle =	{7th Symposium on Foundations of Responsible Computing (FORC 2026)},
  pages =	{14:1--14:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-419-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{368},
  editor =	{Lin, Huijia (Rachel)},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2026.14},
  URN =		{urn:nbn:de:0030-drops-259872},
  doi =		{10.4230/LIPIcs.FORC.2026.14},
  annote =	{Keywords: Bayesian Persuasion, Fair Division, Submodular Optimization}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.34
Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The k-median and k-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the k-median (resp. k-means) problem is to find k representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a (1+ε)-factor approximation in low-dimensional Euclidean metric for both the k-median and k-means problems in near-linear time 2^{(1/ε)^O(d²)} n ⋅ polylog(n) (where d is the dimension and n is the number of input points). We improve this running time to 2^{O(1/ε)^{d-1}} ⋅ n ⋅ polylog(n), and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no 2^o(1/ε^{d-1}) n^O(1) algorithm achieving a (1+ε)-approximation for k-means.
Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.34,
  author =	{Cohen-Addad, Vincent and Karthik C. S. and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.34},
  URN =		{urn:nbn:de:0030-drops-258404},
  doi =		{10.4230/LIPIcs.SoCG.2026.34},
  annote =	{Keywords: k-means clustering, k-median clustering, Euclidean space, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.46
A Branch-And-Bound Algorithm for the Traveling Salesman Problem with Difficult Neighborhoods

Authors: Sándor P. Fekete, Rouven Kniep, Dominik Krupke, and Michael Perk

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The Traveling Salesman Problem with Neighborhoods (TSPN) generalizes the classical Traveling Salesman Problem (TSP) by requiring a tour to visit a set of polygonal regions rather than fixed points, a natural goal that arises in various applications. While the geometric TSP allows arbitrarily close approximation and provably optimal solutions for benchmark instances of significant size, the TSPN is considerably more challenging, both in theory (due to APX-hardness) and practice, for which only benchmark instances up to 16 regions have been solved to optimality. Here we present a branch-and-bound algorithm that combines a spectrum of geometry-based filters (for reducing the number of considered sequences) with Second-Order Cone Programs (SOCP) (for computing optimal tours for a given permutation of neighborhoods). This allows us to solve larger polygonal TSPN instances than before to within an optimality tolerance of 0.1%; moreover, while previous work (both in theory and practice) relied on relatively benign neighborhoods, we can handle non-convex, non-simple neighborhoods of different sizes. In experiments on 490 benchmark instances with up to 50 polygons each, our method achieves a 99.6% optimality rate within 300s, with the remaining two instances solved within 595s. For 68 larger instances of size n = 60, our method still allows solving 86.8% of instances to optimality within 900s, leaving only 3 of the instances with optimality gaps above 3%, with the maximum being 5.53%.
Cite as

Sándor P. Fekete, Rouven Kniep, Dominik Krupke, and Michael Perk. A Branch-And-Bound Algorithm for the Traveling Salesman Problem with Difficult Neighborhoods. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.46,
  author =	{Fekete, S\'{a}ndor P. and Kniep, Rouven and Krupke, Dominik and Perk, Michael},
  title =	{{A Branch-And-Bound Algorithm for the Traveling Salesman Problem with Difficult Neighborhoods}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{46:1--46:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.46},
  URN =		{urn:nbn:de:0030-drops-258529},
  doi =		{10.4230/LIPIcs.SoCG.2026.46},
  annote =	{Keywords: Geometric optimization, geometric covering, TSP with neighborhoods, exact algorithms, algorithm engineering}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.64
Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree

Authors: Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh, and Geert van Wordragen

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We give an approximation scheme for the TSP in d-dimensional hyperbolic space that has optimal dependence on ε under Gap-ETH. For any fixed dimension d ≥ 2 and for any ε > 0 our randomized algorithm gives a (1+ε)-approximation in time 2^O(1/ε^{d-1}) n^{1+o(1)}. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time. Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.
Cite as

Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh, and Geert van Wordragen. Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2026.64,
  author =	{Kisfaludi-Bak, S\'{a}ndor and Odak, Saeed and Singh, Satyam and van Wordragen, Geert},
  title =	{{Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{64:1--64:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.64},
  URN =		{urn:nbn:de:0030-drops-258710},
  doi =		{10.4230/LIPIcs.SoCG.2026.64},
  annote =	{Keywords: Hyperbolic traveling salesman problem, TSP, Hyperbolic Steiner tree problem, Approximation scheme, Banyan, Hyperbolic geometry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.91
The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks

Authors: Jonathan Richard Shewchuk and Sagnik Bhattacharya

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
A linear neural network computes a linear transformation of its input vector. Given a fully-connected linear network, the set of all weight vectors for which the network computes the same linear transformation is an algebraic variety in weight space, called a fiber under the matrix multiplication map. Sometimes this variety is a manifold, but usually not. The rank stratification of a fiber is a natural partition of the fiber into manifolds of various dimensions called strata. We characterize how these strata are connected to each other. They satisfy the frontier condition: if a stratum intersects the closure of another stratum, then the former stratum is a subset of the closure of the latter stratum. This subset relationship can be expressed as a partial order with a single minimal element. Our main result describes the relationship between this partial order and the ranks of certain matrices in the network. Each stratum represents a different pattern of information flow through the network, expressed as a barcode. Connections among the strata are best understood through simple transformations of the barcodes called barcode moves.
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Jonathan Richard Shewchuk and Sagnik Bhattacharya. The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 91:1-91:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{shewchuk_et_al:LIPIcs.SoCG.2026.91,
  author =	{Shewchuk, Jonathan Richard and Bhattacharya, Sagnik},
  title =	{{The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{91:1--91:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.91},
  URN =		{urn:nbn:de:0030-drops-258971},
  doi =		{10.4230/LIPIcs.SoCG.2026.91},
  annote =	{Keywords: Linear neural network, real algebraic variety, stratification, multilinear algebra, product of matrices, persistence barcode, real algebraic geometry, discrete geometry}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.41
On the Complexity of Computing Strahler Numbers

Authors: Moses Ganardi and Markus Lohrey

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
It is shown that the problem of computing the Strahler number of a binary tree given as a term is complete for the circuit complexity class uniform NC¹. For several variants, where the binary tree is given by a pointer structure or in a succinct form by a directed acyclic graph or a tree straight-line program, the complexity of computing the Strahler number is determined as well. The problem, whether a given context-free grammar in Chomsky normal form produces a derivation tree (resp., an acyclic derivation tree), whose Strahler number is at least a given number k is shown to be P-complete (resp., PSPACE-complete).
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Moses Ganardi and Markus Lohrey. On the Complexity of Computing Strahler Numbers. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ganardi_et_al:LIPIcs.STACS.2026.41,
  author =	{Ganardi, Moses and Lohrey, Markus},
  title =	{{On the Complexity of Computing Strahler Numbers}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{41:1--41:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.41},
  URN =		{urn:nbn:de:0030-drops-255301},
  doi =		{10.4230/LIPIcs.STACS.2026.41},
  annote =	{Keywords: Strahler number, circuit complexity classes, context-free grammars}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.25
Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Authors: Marek Černý and Tim Seppelt

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Two graphs G and H are homomorphism indistinguishable over a graph class ℱ if they admit the same number of homomorphisms from every graph F ∈ ℱ. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems HomInd(ℱ) of deciding homomorphism indistinguishability over ℱ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of HomInd(ℱ), Seppelt (MFCS 2024) showed that HomInd(ℱ) is in randomised polynomial time for every graph class ℱ of bounded treewidth that can be defined in counting monadic second-order logic CMSO₂. We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in P. For CMSO₂-definable graph classes ℱ of bounded pathwidth, we improve the previous complexity upper bound for HomInd(ℱ) from P to C_ = L and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as C_ = L-complete.
Cite as

Marek Černý and Tim Seppelt. Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cerny_et_al:LIPIcs.STACS.2026.25,
  author =	{\v{C}ern\'{y}, Marek and Seppelt, Tim},
  title =	{{Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.25},
  URN =		{urn:nbn:de:0030-drops-255144},
  doi =		{10.4230/LIPIcs.STACS.2026.25},
  annote =	{Keywords: treewidth, Courcelle’s theorem, logspace, multiplicity automata, polynomial identity testing}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.37
New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String

Authors: Lijie Chen, Yang Hu, and Hanlin Ren

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The algebrization barrier, proposed by Aaronson and Wigderson (STOC '08, ToCT '09), captures the limitations of many complexity-theoretic techniques based on arithmetization. Notably, several circuit lower bounds that overcome the relativization barrier (Buhrman-Fortnow-Thierauf, CCC '98; Vinodchandran, TCS '05; Santhanam, STOC '07, SICOMP '09) remain subject to the algebrization barrier. In this work, we establish several new algebrization barriers to circuit lower bounds by studying the communication complexity of the following problem, called XOR-Missing-String: For m < 2^{n/2}, Alice gets a list of m strings x₁, … , x_m ∈ {0, 1}ⁿ, Bob gets a list of m strings y₁, … , y_m ∈ {0, 1}ⁿ, and the goal is to output a string s ∈ {0, 1}ⁿ that is not equal to x_i⊕ y_j for any i, j ∈ [m]. 1) We construct an oracle A₁ and its multilinear extension A₁̃ such that PostBPE^{A₁̃} has linear-size A₁-oracle circuits on infinitely many input lengths. That is, proving PostBPE ̸ ⊆ i.o.- SIZE[O(n)] requires non-algebrizing techniques. This barrier follows from a PostBPP communication lower bound for XOR-Missing-String. This is in contrast to the well-known algebrizing lower bound MA_E (⊆ PostBPE) ̸ ⊆ P/_poly. 2) We construct an oracle A₂ and its multilinear extension A₂̃ such that BPE^{A₂̃} has linear-size A₂-oracle circuits on all input lengths. Previously, a similar barrier was demonstrated by Aaronson and Wigderson, but in their result, A₂̃ is only a multiquadratic extension of A₂. Our results show that communication complexity is more useful than previously thought for proving algebrization barriers, as Aaronson and Wigderson wrote that communication-based barriers were "more contrived". This serves as an example of how XOR-Missing-String forms new connections between communication lower bounds and algebrization barriers. 3) Finally, we study algebrization barriers to circuit lower bounds for MA_E. Buhrman, Fortnow, and Thierauf proved a sub-half-exponential circuit lower bound for MA_E via algebrizing techniques. Toward understanding whether the half-exponential bound can be improved, we define a natural subclass of MA_E that includes their hard MA_E language, and prove the following result: For every super-half-exponential function h(n), we construct an oracle A₃ and its multilinear extension A₃̃ such that this natural subclass of MA_E^{A₃̃} has h(n)-size A₃-oracle circuits on all input lengths. This suggests that half-exponential might be the correct barrier for MA_E circuit lower bounds w.r.t. algebrizing techniques.
Cite as

Lijie Chen, Yang Hu, and Hanlin Ren. New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2026.37,
  author =	{Chen, Lijie and Hu, Yang and Ren, Hanlin},
  title =	{{New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{37:1--37:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.37},
  URN =		{urn:nbn:de:0030-drops-253246},
  doi =		{10.4230/LIPIcs.ITCS.2026.37},
  annote =	{Keywords: circuit lower bound, algebrization barrier, missing string, communication complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.95
Identity Testing for Circuits with Exponentiation Gates

Authors: Jiatu Li and Mengdi Wu

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Motivated by practical applications in the design of optimization compilers for neural networks, we initiated the study of identity testing problems for arithmetic circuits augmented with exponentiation gates that compute the real function x↦ e^x. These circuits compute real functions of form P(→x)/P'(→x), where both P(→x) and P'(→x) are exponential polynomials ∑_{i = 1}^k f_i(→x)⋅ exp((g_i(→x))/(h_i(→x))), for polynomials f_i(→x),g_i(→x), and h_i(→x). We formalize a black-box query model over finite fields for this class of circuits, which is mathematical simple and reflects constraints faced by real-world neural network compilers. We proved that a simple and efficient randomized identity testing algorithm achieves perfect completeness and non-trivial soundness. Concurrent with our work, the algorithm has been implemented in the optimization compiler Mirage by Wu et al. (OSDI 2025), demonstrating promising empirical performance in both efficiency and soundness error. Finally, we propose a number-theoretic conjecture under which our algorithm is sound with high probability.
Cite as

Jiatu Li and Mengdi Wu. Identity Testing for Circuits with Exponentiation Gates. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 95:1-95:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{li_et_al:LIPIcs.ITCS.2026.95,
  author =	{Li, Jiatu and Wu, Mengdi},
  title =	{{Identity Testing for Circuits with Exponentiation Gates}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{95:1--95:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.95},
  URN =		{urn:nbn:de:0030-drops-253821},
  doi =		{10.4230/LIPIcs.ITCS.2026.95},
  annote =	{Keywords: Polynomial Identity Testing, Exponential Polynomials}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.42
Time and Space Efficient Deterministic List Decoding

Authors: Joshua Cook and Dana Moshkovitz

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Error correcting codes encode messages by codewords in such a way that even if some of the codeword is corrupted, the message can be decoded. Typical decoding algorithms for error correcting codes either use linear space or quadratic time. A natural question is whether codes can be decoded in near-linear time and sub-linear space simultaneously. A recent result by Cook and Moshkovitz gave efficient decoders that can uniquely decode Reed-Muller and other codes from a constant fraction (less than half) of corruption. In this work, we address the problem of list decoding in near-linear time and sub-linear space. In the list decoding setting, most of the codeword is corrupted, and one wants to output a short list of potential messages that contains the true message. For any constants γ, τ > 0, we give decoders for Reed-Muller codes that can decode from 1-γ fraction of corruptions in time n^{1+τ} and space n^{τ}. Our decoders work by extending the iterative correction technique of Cook and Moshkovitz. However, that technique, which gradually decreases the number of corruptions in the message, was tailored to the unique decoding setting. We first identify an intermediate problem, codewords list recovery, for which we can make iterative correction work. We then show how to reduce general list decoding to the codewords list recovery problem in efficient time and space. The reduction relies on local correction and testing. In the codewords list recovery problem, the input consists of n unordered lists containing exactly the symbols from L codewords, where a small fraction of the lists is corrupted. The goal is to find the L codewords. In addition, we prove that any linear code with time-space efficient encoding or decoding must be local, in the sense that the codewords satisfy a local linear constraint. This rules out codes like Reed-Solomon from having time-space efficient encoding or decoding.
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Joshua Cook and Dana Moshkovitz. Time and Space Efficient Deterministic List Decoding. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 42:1-42:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cook_et_al:LIPIcs.ITCS.2026.42,
  author =	{Cook, Joshua and Moshkovitz, Dana},
  title =	{{Time and Space Efficient Deterministic List Decoding}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{42:1--42:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.42},
  URN =		{urn:nbn:de:0030-drops-253292},
  doi =		{10.4230/LIPIcs.ITCS.2026.42},
  annote =	{Keywords: Reed-Muller code, local correction, local testing}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.23
Decoding Balanced Linear Codes with Preprocessing

Authors: Andrej Bogdanov, Rohit Chatterjee, Yunqi Li, and Prashant Nalini Vasudevan

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Prange’s information set algorithm is a well-known decoding algorithm for linear codes. It decodes corrupted codewords of most 𝔽₂-linear codes C of message length n up to relative error rate O(log n / n) in poly(n) time. We show that the error rate can be improved to O((log n)² / n), provided: (1) the decoder has access to a polynomial-length advice string that depends on C only, and (2) C is n^{-Ω(1)}-balanced. As a consequence we improve the error tolerance in decoding random linear codes if inefficient preprocessing of the code is allowed. This reveals potential vulnerabilities in cryptographic applications of Learning Noisy Parities with low noise rate. Our main technical result is that the Hamming weight of Hw, where the rows of H are a random sample of short dual codewords, measures the proximity of a received word w to the code in the regime of interest. Given such H as advice, our algorithm corrects errors by locally minimizing this measure. We show that for most codes, the error rate tolerated by our decoder is asymptotically optimal among all algorithms whose decision is based on thresholding Hw for an arbitrary polynomial-size advice matrix H.
Cite as

Andrej Bogdanov, Rohit Chatterjee, Yunqi Li, and Prashant Nalini Vasudevan. Decoding Balanced Linear Codes with Preprocessing. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 23:1-23:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bogdanov_et_al:LIPIcs.ITCS.2026.23,
  author =	{Bogdanov, Andrej and Chatterjee, Rohit and Li, Yunqi and Vasudevan, Prashant Nalini},
  title =	{{Decoding Balanced Linear Codes with Preprocessing}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{23:1--23:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.23},
  URN =		{urn:nbn:de:0030-drops-253107},
  doi =		{10.4230/LIPIcs.ITCS.2026.23},
  annote =	{Keywords: Linear codes, nearest codeword problem, learning parity with noise}
}
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